The angular displacement is generated when an object moves along a path or path having circumferentially. It is different from displacement; While angular displacement measures the angle traveled, displacement measures distance.
Two ways can be used to calculate the angular displacement of an object moving along a circumference: if the initial and final angle is known, then the angular displacement will be the subtraction between the final angle and the initial angle.
Graphical representation of an angular displacement
If the length of the displacement (length of the arc of circumference traveled) and the radius of the circumference are known, then the angular displacement is given by θ = l / r.
Formulas
To obtain the formulas described above, the following images can be observed:
The first shows why the angular displacement is equal to the subtraction of the final angle minus the initial angle.
In the second image is the formula for the length of an arc of circumference. Therefore, solving for θ the formula described at the beginning is obtained.
Exercises
Below are some exercises where the definition of angular displacement should be applied and where the formulas described above are used.
First exercise
Juan has run a distance of 35 meters on a circular athletics track whose radius is equal to 7 meters. Find the angular displacement that Juan has made.
Solution
Since the distance of the arc traveled and the radius of the circumference are known, the second formula can be applied to know the angular displacement made by Juan. Using the formula described above, we have θ = 35/7 = 5 radians.
Second exercise
If Mario has traveled half a circular race track in his vehicle, what is the angular displacement that Mario has made?
Solution
In this exercise the first formula will be applied. Since Mario is known to have covered the middle of the track, it can be assumed that he started the race at the 0 ° angle and when he reached the middle of the circumference he had traveled 180 °. Therefore, the answer is 180 ° -0 ° = 180 ° = π radians.
Third exercise
Maria has a circular pool. Your dog runs around the pool for a distance of 18 meters. If the radius of the pool is 3 meters, what is the angular displacement made by Maria's pet?
Solution
Since the pool is circular and the radius of the pool is known, the second formula can be used.
It is known that the radius is equal to 3 meters, and the distance traveled by the pet is equal to 18 meters. Therefore, the realized angular displacement is equal to θ = 18/3 = 6 radians.
References
- Basto, JR (2014). Mathematics 3: Basic Analytical Geometry. Grupo Editorial Patria.
- Billstein, R., Libeskind, S., & Lott, JW (2013). Mathematics: A Problem Solving Approach for Elementary Education Teachers. López Mateos Editors.
- Bult, B., & Hobbs, D. (2001). Lexicon of mathematics (illustrated ed.). (FP Cadena, Trad.) AKAL Editions.
- Callejo, I., Aguilera, M., Martínez, L., & Aldea, CC (1986). Maths. Geometry. Reform of the upper cycle of the EGB Ministry of Education.
- Schneider, W., & Sappert, D. (1990). Practical manual of technical drawing: introduction to the fundamentals of industrial technical drawing. Reverte.
- Thomas, GB, & Weir, MD (2006). Calculation: several variables. Pearson Education.